Cubic critical

Average Error: 53.94% → 16.61%

Time: 22.7s

Precision: binary64

Cost: 7624

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e+131)
   (/ b (* a -1.5))
   (if (<= b 3e-135)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+131) {
		tmp = b / (a * -1.5);
	} else if (b <= 3e-135) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d+131)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 3d-135) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+131) {
		tmp = b / (a * -1.5);
	} else if (b <= 3e-135) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -2.4e+131:
		tmp = b / (a * -1.5)
	elif b <= 3e-135:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e+131)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 3e-135)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e+131)
		tmp = b / (a * -1.5);
	elseif (b <= 3e-135)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -2.4e+131], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-135], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3999999999999999e131

   1. Initial program 88.68

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 88.69

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof

      [Start] 88.68	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 88.68	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 88.68	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 88.68	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 88.68	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 88.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 88.7	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 88.68	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 88.68	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 88.68	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 88.68	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 88.68	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 88.68	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 88.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 88.7	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around -inf 17.58

      \[\leadsto \color{blue}{\left(-1.5 \cdot \frac{c \cdot a}{b} + 2 \cdot b\right)} \cdot \frac{-0.3333333333333333}{a} \]

   4. Taylor expanded in c around 0 5.06

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   5. Simplified 5.07

      \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
      Proof

      [Start] 5.06	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 5	\[ \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]
      metadata-eval [<=] 5	\[ \frac{\color{blue}{\left(-0.3333333333333333 \cdot 2\right)} \cdot b}{a} \]
      associate-*r* [<=] 5	\[ \frac{\color{blue}{-0.3333333333333333 \cdot \left(2 \cdot b\right)}}{a} \]
      *-commutative [<=] 5	\[ \frac{-0.3333333333333333 \cdot \color{blue}{\left(b \cdot 2\right)}}{a} \]
      associate-*l/ [<=] 5.07	\[ \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot 2\right)} \]
      *-commutative [<=] 5.07	\[ \color{blue}{\left(b \cdot 2\right) \cdot \frac{-0.3333333333333333}{a}} \]
      associate-*l* [=>] 5.07	\[ \color{blue}{b \cdot \left(2 \cdot \frac{-0.3333333333333333}{a}\right)} \]
      associate-*r/ [=>] 5.07	\[ b \cdot \color{blue}{\frac{2 \cdot -0.3333333333333333}{a}} \]
      metadata-eval [=>] 5.07	\[ b \cdot \frac{\color{blue}{-0.6666666666666666}}{a} \]

   6. Applied egg-rr 4.89

      \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.3999999999999999e131 < b < 3.00000000000000012e-135

   1. Initial program 17.37

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   if 3.00000000000000012e-135 < b

   1. Initial program 79.18

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 79.23

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
      Proof

      [Start] 79.18	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 79.18	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 79.18	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 79.18	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [<=] 79.18	\[ \frac{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}}{-1 \cdot \left(3 \cdot a\right)} \]
      times-frac [=>] 79.19	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-*r/ [=>] 79.18	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot -1}{3 \cdot a}} \]

   3. Applied egg-rr 72.05

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right) - b\right)}}}{3 \cdot a} \]

   4. Taylor expanded in b around inf 100

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]

   5. Simplified 19.2

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
      Proof

      [Start] 100	\[ 0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]
      associate-*r/ [=>] 100	\[ \color{blue}{\frac{0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} \]
      *-commutative [=>] 100	\[ \frac{0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} \]
      associate-*r* [=>] 100	\[ \frac{\color{blue}{\left(0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} \]
      unpow2 [=>] 100	\[ \frac{\left(0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} \]
      rem-square-sqrt [=>] 19.2	\[ \frac{\left(0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} \]
      metadata-eval [=>] 19.2	\[ \frac{\color{blue}{-0.5} \cdot c}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 16.61

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.67%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Error	21.96%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Error	21.98%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Error	21.96%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-90}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Error	21.92%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Error	35.71%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-198}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	35.71%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{-197}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	35.65%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	35.65%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-198}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10
Error	61.75%
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Alternative 11
Error	87.84%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))